In linear control, for mathematical simplicity, it is typically assumed that noise is additive. Typical dynamics look like

where

are the state, input, and noise, respectively. In particular, note that the noise does not multiply the state or the input.

For human and animal movements, however, the noise is not additive. The inputs to muscles are motor neurons. As the motor neuron firing rate increases, it becomes more variable. In other words, larger inputs lead to more noise.

Linear dynamics with signal-dependent noise, as just described, can be modeled as

While the math gets more complicated when the noise multiplies the input, such models can predict numerous qualitative features of natural movements. For example, according to Fitts’s law, higher precision movements require more time. Under additive noise, fast movements will be more precise than slow movements because less noise is injected into the system. Under the signal-dependent noise model, the large inputs associated with fast movements will lead to large noises, and hence imprecise movements. Thus, the two models lead to opposite predictions, with the signal-dependent noise model being more consistent with the data.

When learning about this model, I wrote some basic code for computing minimum variance controllers, which can be found here. More general code is given here.